Question

Va rog frumos ajutatima si pe mine la ecuatia aceasta!! 1/1*2+1/2*3+....+1/x(x+1)=2010/2011

Answer 1

1/1*2=2-1/1*2=2/1*2-1/1*2
1/2*3=3-2/2*3=3/2*3-2/2*3
.....
1/x(x+1)=(x+1)-x/x(x+1)=x+1/x(x+1)-x/x(x+1)
Suma devine:
2/1*2-1/1*2+3/2*3-2/2*3+....+(x+1)/x(x+1)-x/x(x+1)
Efectuam simplificarile in fiecare fractie si obtinem:
1/1-1/2+1/2-1/3+1/3-...+1/x-1/x+1
-1/2 se reduce cu +1/2, -1/3 se reduce cu +1/3...-1/x se reduce cu +1/x
Obtinem:
1/1-1/(x+1)=2010/2011
Aducem la acelasi numitor si obtinem: 
x+1-1/x+1=2010/2011, deci x/x+1=2010/2011, ceea ce inseamna ca x=2010